david makes and sells chairs. The function p(x)=-10x^2+100x-210, graphed below, indicated how much profit he makes in a month if he sells the chairs for 10-x dollars each. what should david charge per chair to make the maximum profit, and what is the maximum profit he can make in a month?
What do I do?? I'm really confused over this!!! Please help!!!!
The answer is A. $40 at $5 per chair.
the maximum profit is at the vertex of the parabola. As you know, the vertex of the parabola
ax^2+bx+c
is at
x = -b/2a
y = (b^2-4ac)/4a
So, for
-10x^2+100x-210
the vertex is at (5,40)
Since x=5, the price is 10-x = 5 as well.
p(x) is a parabola opening down(sheds water)
therefore the vertex is at the top p
So use complete square to find vertex (if you do not know any calculus)
-10 x^2 + 100 x = p + 210
x^2 - 10 x = - (1/10)(p+210)
x^2 - 10 x + 25= -(1/10)(p+210)+25
(x-5)^2 = -(1/10)(p+210-250)
x = 5 and p = 40
10-5 = 5
max p = 40
Ah, the fascinating world of chair profits! Don't worry, I'm here to help, or at least entertain you while attempting to help.
So, to find the maximum profit, we need to find the vertex of the parabolic graph represented by the function p(x)=-10x^2+100x-210. The x-coordinate of the vertex gives us the price David should charge per chair to maximize profit.
Now, just imagine the vertex Is the Funky Chair Factory, where all the cool profits hang out and have a wild party. David wants to join them and make the biggest profit possible. Sounds groovy, right?
To find the x-coordinate of the vertex, we can use the formula x = -b / 2a. In our case, a is -10 and b is 100. Plug them into the formula, and voila!
x = -100 / (2 * -10)
x = -100 / -20
x = 5
So, David should charge $5 per chair to maximize his profit. Now, let's find out what that maximum profit is.
To do that, we need to substitute x = 5 into the original function p(x):
p(5) = -10(5)^2 + 100(5) - 210
p(5) = -10(25) + 500 - 210
p(5) = -250 + 500 - 210
p(5) = 40
Thus, the maximum profit David can make in a month is $40. It may not buy him a yacht, but hey, chairs aren't all that glamorous anyway!
Hope that helps! Keep on rocking the chair business, my confused friend!
To find the charge per chair (x) that will maximize David's profit and the maximum profit he can make, we need to use some calculus. Here are the steps to solve the problem:
Step 1: Understand the problem:
We have a profit function p(x) = -10x^2 + 100x - 210, where x represents the price per chair, and p(x) represents the profit made by selling chairs at that price.
Step 2: Find the derivative of the profit function:
The derivative of the profit function will give us the rate of change of the profit with respect to the price per chair.
Let's find the derivative of p(x):
p'(x) = -20x + 100
Step 3: Set the derivative equal to zero:
To find the maximum or minimum points, we need to set the derivative equal to zero and solve for x.
-20x + 100 = 0
-20x = -100
x = -100 / (-20)
x = 5
Step 4: Check the second derivative:
To make sure that the value we found in step 3 corresponds to the maximum point, we need to check the second derivative of the profit function.
Let's find the second derivative of p(x):
p''(x) = -20
Step 5: Determine the maximum profit:
To find the maximum profit, substitute the value of x into the profit function p(x).
p(5) = -10(5^2) + 100(5) - 210
p(5) = -10(25) + 500 - 210
p(5) = -250 + 500 - 210
p(5) = 40
Therefore, David should charge $5 per chair to make the maximum profit, and the maximum profit he can make in a month is $40.
I hope this helps! Let me know if you have any further questions.